Solve for p p^4-p^2-20=0

$p^{4} - p^{2} - 20 = 0$

Step 1

Substitute $u = p^{2}$ into the equation. This will make the quadratic formula easy to use.

$u^{2} - u - 20 = 0$

$u = p^{2}$

Step 2

Factor $u^{2} - u - 20$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 20$ and whose sum is $- 1$ .

$- 5, 4$

Write the factored form using these integers.

$(u - 5) (u + 4) = 0$

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$u - 5 = 0$

$u + 4 = 0$

Step 4

Set $u - 5$ equal to $0$ and solve for $u$ .

Set $u - 5$ equal to $0$ .

$u - 5 = 0$

Add $5$ to both sides of the equation.

$u = 5$

Step 5

Set $u + 4$ equal to $0$ and solve for $u$ .

Set $u + 4$ equal to $0$ .

$u + 4 = 0$

Subtract $4$ from both sides of the equation.

$u = - 4$

Step 6

The final solution is all the values that make $(u - 5) (u + 4) = 0$ true.

$u = 5, - 4$

Step 7

Substitute the real value of $u = p^{2}$ back into the solved equation.

$p^{2} = 5$

${(p^{2})}^{1} = - 4$

Step 8

Solve the first equation for $p$ .

$p^{2} = 5$

Step 9

Solve the equation for $p$ .

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$p = \pm \sqrt{5}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$p = \sqrt{5}$

Next, use the negative value of the $\pm$ to find the second solution.

$p = - \sqrt{5}$

The complete solution is the result of both the positive and negative portions of the solution.

$p = \sqrt{5}, - \sqrt{5}$

Step 10

Solve the second equation for $p$ .

${(p^{2})}^{1} = - 4$

Step 11

Solve the equation for $p$ .

Remove parentheses.

$p^{2} = - 4$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$p = \pm \sqrt{- 4}$

Simplify $\pm \sqrt{- 4}$ .

Rewrite $- 4$ as $- 1 (4)$ .

$p = \pm \sqrt{- 1 (4)}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$p = \pm \sqrt{- 1} \cdot \sqrt{4}$

Rewrite $\sqrt{- 1}$ as $i$ .

$p = \pm i \cdot \sqrt{4}$

Rewrite $4$ as $2^{2}$ .

$p = \pm i \cdot \sqrt{2^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$p = \pm i \cdot 2$

Move $2$ to the left of $i$ .

$p = \pm 2 i$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$p = 2 i$

Next, use the negative value of the $\pm$ to find the second solution.

$p = - 2 i$

The complete solution is the result of both the positive and negative portions of the solution.

$p = 2 i, - 2 i$

Step 12

The solution to $p^{4} - p^{2} - 20 = 0$ is $p = \sqrt{5}, - \sqrt{5}, 2 i, - 2 i$ .

$p = \sqrt{5}, - \sqrt{5}, 2 i, - 2 i$

Solve for p p^4-p^2-20=0

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